The symbol is located at the base line. Probably you want to center it around the math axis (like +, ×, ...).

The height of the symbol is not defined. The example uses the height of H.

Subscripts and superscripts are not arguments, thus the rule that curly braces can be omitted, if there is only one token does not apply. It only works, if the token expands to a math symbol or subformula. The curly braces work, because they form a subformula.

The example uses \resizebox, because it is more flexible than options to \includegraphics. Then other stuff (tikzpicture, ...) can be used instead of \includegraphics.

The repeated use of \includegraphics can be avoided by using a box register. The image is stored in box register \mysymbolbox and reused in macro \mysymbol. This also avoids that the .log file is flooded with information data about the image mysymbol.